Question

if a data value in a normal distribution has a negative z-score, which of the following must be true?
the data value must be negative.
the data value must be positive.
the data value must be less than the mean.
the data value must be greater than the mean.

Explanation:

Step1: Recall Z - score formula

The formula for the z - score is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the data value, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation (and \(\sigma>0\) since it is a measure of spread).

Step2: Analyze negative z - score

If \(z<0\), then from the formula \(z = \frac{x-\mu}{\sigma}\), since \(\sigma>0\), we can multiply both sides of the inequality \(z<0\) by \(\sigma\) (which doesn't change the direction of the inequality) to get \(x - \mu<0\). Then, by adding \(\mu\) to both sides of the inequality \(x-\mu < 0\), we obtain \(x<\mu\). This means that the data value \(x\) is less than the mean \(\mu\).

Now let's analyze the other options:

• Option "The data value must be negative": The z - score depends on the difference between the data value and the mean relative to the standard deviation. For example, if the mean \(\mu = 10\) and \(\sigma=5\), a data value \(x = 8\) has a z - score \(z=\frac{8 - 10}{5}=\frac{- 2}{5}=- 0.4\) (negative z - score) but the data value \(x = 8\) is positive. So this option is incorrect.
• Option "The data value must be positive": Using the same example as above, if \(\mu = 3\) and \(\sigma = 1\), a data value \(x=1\) has a z - score \(z=\frac{1 - 3}{1}=-2\) (negative z - score) and the data value \(x = 1\) is positive? No, \(x = 1\) is positive? Wait, no, \(x = 1\) is positive, but if \(\mu=-1\) and \(\sigma = 1\), a data value \(x=-3\) has a z - score \(z=\frac{-3-(-1)}{1}=\frac{-2}{1}=-2\) (negative z - score) and the data value is negative. So a data value with a negative z - score can be positive or negative, so this option is incorrect.
• Option "The data value must be greater than the mean": From our earlier derivation, we found that \(x<\mu\) when \(z < 0\), so this option is incorrect.

Answer:

The data value must be less than the mean. (The option corresponding to "The data value must be less than the mean")